# Lesson 4

Interpreting Functions

These materials, when encountered before Algebra 1, Unit 4, Lesson 4 support success in that lesson.

## 4.1: Math Talk: Finding Outputs (10 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for finding outputs of functions given an input. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate functions for given inputs in context.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

Mentally evaluate the output for the input of 3.

\(f(x) = 4\left( x - \frac{1}{2}\right)\)

\(g(x) = 2(6 - x)\)

\(h(x) = \frac{5}{3}x + \frac{1}{3}\)

\(j(x) = 0.2x - 1\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## 4.2: It’s Getting Hotter (15 minutes)

### Activity

In this activity, students connect a function to its meaning in a situation. Students interpret the input and output as well as use the function to find the output for a given input and the input for a given output. In the associated Algebra 1 lesson, students interpret different representations in the context of the problem. This activity supports students by allowing them to focus on one representation, the equation.

### Student Facing

A machine in a laboratory is set to steadily increase the temperature inside. The temperature in degrees Celsius inside the machine after being turned on is a function of time, in seconds, given by the equation \(f(t) = 22 + 1.3t\).

- What does \(f(3)\) mean in this situation?
- Find the value of \(f(3)\) and interpret that value.
- What does the equation \(f(t) = 35\) mean in this situation?
- Solve the equation to find the value of \(t\) for the previous question.
- Write an equation involving \(f\) that represents each of these situations:
- The temperature in the machine 30 seconds after it is turned on.
- The time when the temperature inside the machine is 100 degrees Celsius.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to understand the meaning of the situation and how it is connected to the notation. Select students to share their responses. Ask students:

- “What are the two variables in the problem? Which is the input? Which is the output? (Time and temperature are the variables. Time is the input and temperature is the output.)
- “When you are given a value inside the parentheses of a function, like \(f(8)\), is the value an input or output?” (The value is an input.)

## 4.3: You Charge How Much? (20 minutes)

### Activity

In this activity, students interpret two functions in the same situation. They compare function values for the same input and inputs for the same function values. In the associated Algebra 1 lesson, students interpret functions in context using multiple representations of the functions. Students are supported in this activity by considering only the equation and graphical representations.

### Student Facing

- Which line represents \(y = f(t)\)? Explain how you know.
- The lines intersect at the point \((4,900)\). What does this point mean in this situation?
- Which is greater, \(f(10)\) or \(g(10)\)? What does that mean in this situation?
- Your group has $1,500 to spend on supercomputer time. Which company should your group use?
- Explain or show your reasoning using the equations.
- Explain or show your reasoning using the graph.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to connect the situation to graphical and equation representations of a function. Select students to share their solutions. Ask students:

- “How do the slope and vertical intercept of a linear equation show up on a graph?” (The vertical intercept is where the graph crosses the vertical axis and the slope is how much the graph rises every 1 unit moved to the right.)
- “What is the slope of the line representing the cost of supercomputer time for the second company (the one represented by \(g(t)\))? What does that slope mean in this situation?” (The slope is 150. This means each additional hour of supercomputer time will cost another $150.)
- “If you work for the first company offering supercomputer time, how would you describe your pricing to a customer?” (We charge $500 to use our supercomputer and an additional $100 for every hour you use the supercomputer.)